Existence of Solutions to Hyperbolic Differential Equations in Mathematical Physics

本项目旨在了解爱因斯坦引力场方程解在各种物质场（如电磁场和流体场）影响下的大时间行为。解的渐近行为和散射的研究对数值相对论和物理界有重要意义。作为该项目一部分的数学发展也有助于分析引力波探测器的数据。研究生的培训是该项目的一部分。人们期望，爱因斯坦方程和黑洞稳定性的简化存在性证明将使学生更容易开始数学相对论的研究。本课题研究爱因斯坦方程和其他场方程的全局解的存在性、渐近行为和散射问题。一个目标是研究不同物质场下爱因斯坦方程小解的渐近行为。首席研究员（PI）已经确定了爱因斯坦方程和许多其他物理系统中的非线性项满足的弱零条件，并且理解这如何决定渐近行为是很重要的。物质场的传播速度比光速慢，因此集中在向前光锥的内部。PI发现引力场的一部分由以光速传播的波组成，另一部分是均匀衰减函数。发现物质如何与引力相互作用是很有趣的。该项目正在开发一种新的、更明确的方法，从无穷远处的散射数据往回构建解。利用显式表达式，确定了散射数据必须满足的一个相容条件，该条件是否足以存在是一个有趣的问题。能够从远处计算致密双星系统（如两个黑洞碰撞）发出的引力波形是很重要的。这些波形被用来从引力波探测器的数据中识别这样的系统。调和坐标下的PI公式特别适合于此，并且被他的合作者在数值相对论中使用。该项目的另一个目标是研究像恒星或黑洞这样的大溶液的稳定性。特别是，重要的是简化和推广最近关于黑洞稳定性的工作所涉及的证明。这个项目的方法是首先更好地理解接近黑洞的一般更简单的模型波动方程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is aimed at understanding the large time behavior of solutions of Einstein's equations for the gravitational field under the influence of various matter fields, such as electromagnetism and fluids. The study of the asymptotic behavior and scattering of solutions are useful for the numerical relativity and physics communities. The mathematics developed as part of the project also helps analyze the data from gravitational wave detectors. Graduate students are trained as part of the project. It is expected that the simplified existence proofs for Einstein's equations and for the stability of black holes will make it easier for students to get started on research in mathematical relativity. The project is concerned with the problem of existence, asymptotic behavior, and scattering of global solutions of Einstein's equations and other field equations. One goal is to study the asymptotic behavior of small solutions of Einstein's equations with various matter fields. The Principal Investigator (PI) has identified a weak null condition that the nonlinear terms in Einstein’s equations and many other physical systems satisfy, and it is important to understand how this determines the asymptotic behavior. The matter fields propagate slower than the speed of light and are hence concentrated in the interior of the forward light cone. The PI discovered that the gravitational field has one part consisting of a wave that travels with the speed of light and one interior part that is a homogeneous decaying function. It is interesting to find out how matter interacts with gravity. The project is developing a new, more explicit, way to construct solutions backwards from scattering data at infinity. Using the explicit expressions, a compatibility condition that scattering data has to satisfy was identified and it would be interesting to find out whether it is sufficient for existence. It is important to be able to compute gravitational waveforms emitted from compact binary systems, such as two black holes colliding, from far away. Those waveforms are being used to identify such systems from the data from gravitational wave detectors. The PI’s formulation in harmonic coordinates is particularly suitable for this and is being used by his collaborators in numerical relativity. Another goal of the project is to study the stability of large solutions like stars or black holes. In particular, it is important to simplify and generalize the proofs involved in the recent work on the stability of black holes. The approach in this project is to first gain a better understanding for general simpler model wave equations close to a black hole.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.